In this section three tests on the developed thermal models used to model transformer and cable loss of lifetime are presented. The purpose of this section give the reader some insight in the known accuracy issues and how to interpret the results of this study. First, the IEEE Clause 7 model and the transformer model by Swift et al. is compared, second the performance of the thermal models for underground cables is assessed, followed by a tradeoff between accuracy and computation time for the Olsen et al. cable model.
3.4.1 Transformer Model Comparison
In this section a comparison is made of the IEEE Clause 7 model and the model by Swift et al. presented in Section 3.2.2.
Figure 3.8 presents a simulation comparison for a cyclic load of 2p.u. for 4 hours and 1p.u. for 20 hours, whereper unit is with respect to the rated load of the transformer. We use the
Table 3.4: Transformer Parameters of the Comparison
Symbol Value Description
LR 2500 kVA Rated load of the transformer
PN L,R 16 kW No-load losses (core) at rated load
PL,R 2211 W Load losses (windings) at rated load
∆θT O,R 48 °C Top-oil temperature rise over ambient at rated load
∆θHS|A,R 72.5 °C Winding hottest spot temperature rise over ambient at rated load
Mcc 2829 kg Weight of the core and coil assembly
MT 1030 kg Weight of the tank and fittings directly in contact with the oil
Voil 1227 L Volume of oil in the transformer
parameters of the transformer found in [21], as presented in Table 3.4. In Figure 3.8, θHS is the
modeled the hottest spot temperature andθT O the top oil temperature for the IEEE Clause 7
and Swift et al. model. Figure 3.8 shows that the model of Swift et al. approaches the steady state value quicker than the Clause 7 model, while the same time constants for the oil and the windings have been used for both models. Note that this does not imply that the model by Swift et al. performs better, only that the response is different for the two models.
The IEEE Loading Guide [11] and Swift et al. in [22] note that there are some known accuracy issues with the estimation of the top oil temperature of the Clause 7 model. However, Swift et al. used parameter estimation to fit the model parameters to the transformer in their validation [22], which does not necessarily mean that parameters from a datasheet give accurate results as well for the Swift et al. model. Therefore we choose to only use the IEEE Clause 7 model in this study. 0 2 4 6 8 10 12 14 16 18 20 22 24 40 60 80 100 120 140 160 Time [h] Temp erature [ ° C] θHS IEEE θHS Swift θT O IEEE θT O Swift
Figure 3.8: Simulation Comparison of IEEE Clause 7 and Swift et al. Models
3.4.2 Cable Thermal Model Performance
This section assesses the performance of the thermal models used in this study. To assure adequate accuracy of the cable thermal models, we conducted a test at rating conditions. The rating conditions should result in a nominal conductor temperature of 55°C. The rating conditions are as follows: the current per phase is 200A, the soil resistivity 0.75 Km/W, the soil temperature is 15°C, and the laying depth 0.6 meters. The thermal resistances presented in Table 3.2 are
used for the IEC 60287 model (Section 3.3.3) and for the Olsen et al. single conductor equivalent (SCE) model (Section 3.3.4), which we refer to as the Olsen et al. model from this point.
For the thermal resistances of IEC 60287 in (3.12),T1 is set to the resistance of the insulation,
T2 to zero,T3 to the sum of the resistances of the filler and sheath, and T4 as the resistance of the soil. Note thatT2 toT4 are multiplied by the number of conductors, whereT1 is not. This is different from the Olsen et al. model, in which effectively speaking all resistances are multiplied by the number of conductors.
Model Conductor [°C] Jacket [°C]
IEC 60287 49.36 36.51
Olsen et al. 52.56 36.51
Expected 55.00 N/A
Table 3.5: Cable Thermal Model Performance
Table 3.5 presents he resulting temperatures of the test at rating conditions. For the Olsen et al. model the steady state value is given, as defined θi(∞) in (3.16). The conductor temperature
is the maximum conductor temperature of the cable, and the jacket temperature or sheath temperature is the temperature of the outer sheath of the cable which touches the soil.
The difference of the conductor temperature of Olsen et al. with the expected temperature is relatively small, taking the inaccuracy into account of manual calculation of the thermal resistances based on the dimensions of cable in the datasheet. The IEC 60287 model however, has a larger deviation. This is probably caused by a correction that is required for the insulation resistanceT1, as the so-calledgeometric factor used in [34]. Because IEC 60287 is not available to us, we use the steady state temperature of the Olsen et al. model instead of the temperature from the IEC 60287 model from this point.
When the load changes each time interval, the dynamic and steady state temperatures are often quite different. An example of this difference is presented in Figure 3.9. Here, the conductor temperatureθcand jacket temperature θj of the Olsen et al. model are plotted for a duty cycle
load of 200A for 16 hours and 0A for 4 hours under the same conditions as the test above. Note that even at the end of the peak from t=4h to t=20h the dynamic temperature does not reach the steady state temperature. This means that the thermal inertia of the cable and soil has a positive effect in decreasing the maximum temperature of the soil. This effect can be modeled using the Olsen et al. model.
In this section we showed the known issues of the thermal models for cables. Furthermore, we chose to use the steady state temperatures of the Olsen et al. instead of the IEC 60287 temperatures in the remainder of this study. Moreover, we showed the effect of the thermal inertia of underground cables of decreasing the maximum conductor temperature.
3.4.3 Cable Model Accuracy vs. Computation Time
This section presents a trade-off between accuracy of the Olsen et al. model and the required computation time of the model.
To determine an adequate amount of rings for the Olsen et al. model, we model a cable and vary the number of rings of the soil from 10, 25, 50, 75, 100, and 200. Only the number of rings of the soil are varied, because the time constants (τi =TiCi) of the rings of the soil are large
compared to the time constants of the rings modeling the cable itself. It is assumed that the model accurately models the conductor temperature for 200 rings. However, we consider the computation time for 200 rings too large.
0 2 4 6 8 10 12 14 16 18 20 22 24 20 30 40 50 60 Time [h] Temp erature [ ° C]
θcSteady State θc Dyn. θj Steady State θj Dyn.
Figure 3.9: A Comparison of Olsen et al. Steady State and Dynamic Ratings
Rings θmax [°C] ∆θmax Comp. Time [ms] Speed Up
200 43.94 0 11220 1.00 100 43.76 0.18 5355 2.10 75 43.64 0.30 4355 2.58 50 43.37 0.57 3689 3.04 25 42.41 1.53 2341 4.79 10 38.69 5.25 1754 6.40
Table 3.6: Effect of the Number of Rings on the Accuracy and Computation Time The corresponding results are presented in Figure 3.10. The colored lines indicate the conductor temperature for different number of rings. The dashed lines indicate the load profile of the cable, the maximum value is equivalent to a current of 200A. Note the effect of increasing the number of rings and that the values for 50 rings and above are relatively close to each other.
Furthermore, Table 3.6 shows the difference in the maximum conductor temperature and computation time with respect to the case of 200 rings. In the table, θmax is the maximum
temperature reached in the simulation, and Speed Up the speed up factor with respect to a simulation using 200 rings. Based on these results we choose the value of 75 for the number of rings. In this case we underestimate the maximum temperature by 0.3°C, while the computation time is 2.58 times faster with respect to 200 rings.
In another attempt to speed up the Olsen et al. model, we tried difference equations to solve the system of differential equations. The fastest implementation however was twice as slow as the approach used by Olsen et al.